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1h
comment Which axioms of ZFC are required to prove the existence of $\aleph _ 0$?
Unless and until you can specify exactly what you mean by $\aleph_0$ there is no point in attempting to answer this question. Usually $\aleph_0$ is synonymous with $\omega$, the least infinite ordinal, or the set of all finite ordinals, or the set of all natural numbers. However you seem to be in a state of disbelief about this, so you must have in mind some other object denoted by $\aleph_0$. It is incumbent on you to describe this object sufficiently well enough that others can answer your question.
4h
revised Probability or Set
rolled back to a previous revision
13h
revised Determining the interior of $([-1, 1]\times[-1, 1])\setminus \{ y \in \mathbb{R}^2 : d((0, 0), y) < 0.25 \} \subseteq \mathbb{R}^2$
added 36 characters in body; edited tags; edited title
15h
comment How does the Soundness Theorem follow from this lemma?
What book (title, author, etc.) is the lemma from?
16h
revised How to find $\lim_{n\to\infty}\left(\frac{\pi^2}{6}-\sum_{k=1}^n\frac{1}{k^2}\right)n$?
added 33 characters in body; edited title
17h
revised Show $\frac{\sin(x)}{x}>\cos(x)$ for $0<x<\pi$ using the Mean Value Theorem
Full statement of the question within the question body. Novel idea, I know.
1d
comment Characterizing uncountable connected topological spaces
I'm not exactly sure what you are asking. Are you asking for topological properties $\Phi$ such that every connected $\Phi$-space of size at least two is uncountable? Would $\Phi = \text{uncountable}$ be acceptable?
1d
comment Possible combinations for 20 character alphanumeric identifier
@AlexM. "...that is made up of 20 alphanumeric characters...." Be made up of: be composed of, be constituted of, consist in, consist of.
1d
comment Possible combinations for 20 character alphanumeric identifier
Question explicitly asks for "20 character" alphanumeric strings, not "at most 20 character" alphanumeric strings.
1d
comment Using chain rule in neural networks
I'm voting to close this question as off-topic because it is a cross-post of a question from Computer Science.
2d
comment Definition of infinite tree in set theory
In set theory a tree is a partial order $( T , \leq )$ where for each $t \in T$ the set $\{ x \in T : x \leq t \}$ is well-ordered by $\leq$. In descriptive set-theory this is restricted to subsets of $A^{< \omega}$ closed under initial segments, with some caveats when dealing with "product trees".
Aug
26
comment Axiomatic proof that all points of an open set are interior points
You want a proof of a definition?
Aug
24
revised Solve this inequality equation with fraction?
added 73 characters in body
Aug
23
comment Winning strategy for graphs (Ehrenfeucht-Fraïssé games)
Might be good to mention which book this is from. (Title, author(s), etc.)
Aug
22
revised Upper and lower bounds for $S(n) = \sum_{i=1}^{2^{n}-1} \frac{1}{i} = 1+\frac{1}{2}+ \cdots +\frac{1}{2^n-1}.$
added 27 characters in body; edited tags; edited title
Aug
22
revised The cardinality of the set all symmetric relations on the set of natural numbers is $\mathfrak{c} = | \mathbb{R} | = 2^{\aleph_0}$
added 66 characters in body; edited tags; edited title
Aug
22
comment Countably closed Boolean algebra of subsets of the real plane,
Which issue of the AMM was this in?
Aug
21
comment Why Newton wanted lines to be generated by continued motion of points rather than by apposition of parts?
Don't cross-post the same questions to multiple sites.
Aug
21
comment If $X$ is a compact $m$-manifold, then $X$ can be imbedded in $\mathbb{R}^N$.
Perhaps include the relevant statement(s) from Munkres (with attribution, of course).
Aug
20
comment Give an example of this strange condition in metric spaces.
See also this question.